Non-solvable Groups whose Character Degree Graph has a Cut-Vertex. I

Abstract

Let G be a finite group. Denoting by $$\textrm{cd}(G)$$ cd ( G ) the set of degrees of the irreducible complex characters of G, we consider the character degree graph of G: this is the (simple undirected) graph whose vertices are the prime divisors of the numbers in $$\textrm{cd}(G)$$ cd ( G ) , and two distinct vertices p, q are adjacent if and only if pq divides some number in $$\textrm{cd}(G)$$ cd ( G ) . In the series of three papers starting with the present one, we analyze the structure of the finite non-solvable groups whose character degree graph possesses a cut-vertex, i.e. a vertex whose removal increases the number of connected components of the graph.

Non-solvable groups whose character degree graph has a cut-vertex. II

“…We claim that ψ does not extend to I. 2 In fact, if the linear character ψ extends to I, then I ′ ∩ B ≤ ker(µ × ϕ 0 ) ∩ B = ker(µ). But, as I = P 2 Q, I ′ = P 3 Q ′ and, using the notation P = P/P 3 , I ′ = Q ′ = Q ′ .…”

“…The character degree graphs possessing a cut-vertex have been studied in [3] and [10] for solvable groups, and they have been fully described for non-solvable groups in [2]. In particular, when G is non-solvable, ∆(G) has a cut-vertex and diameter three if and only if G = J 1 × A, where J 1 is the first Janko group and A is an abelian group ([2, Theorem A(d)]).…”